be considered as that of a sphere; and when a great extent is to be surveyed it may be permitted to determine the situations of the principal points on the supposition that it is a spheroid of revolution. In the latter case the terrestrial meridians are ellipses having their major axes in the plane of the equator, and having a line joining the poles of the latter for a minor axis common to all: the equator and the parallels of latitude are circles, and a diameter of the equator is equal to the major axis of each of the elliptical meridians. The latitude of any point on the earth's surface is expressed (art. 151.) by the angle which a vertical line drawn through the point makes with the plane of the equator; and this is the latitude immediately obtained by the usual processes of practical astronomy. The vertical line just mentioned is a normal to the meridian at the said point.

387. If normals to the earth's surface were supposed to be drawn through any two points on that surface, and to be produced to the celestial sphere, the upper extremity of each would be the geographical zenith of that point: then, if an arc of a great circle were supposed to pass through those two zenith points, and from every point in it, normals to the earth's surface were drawn to meet the latter, the line on that surface, joining the lower extremities of such normals, would constitute the curve of shortest extent between the two points first mentioned; and consequently its rectification would express their true geodetic distance from each other.

If the earth were a sphere this curve line would coincide with an arc of one of its great circles; for all the normals would be in a plane passing through the centre of the earth. If the earth were a spheroid of revolution, and the original points were in the circumference of the equator or of any parallel of latitude, the geodetic line would also be an arc of a circle: if those points were on the periphery of one meridian, all the normals would be in the plane of that meridian, and the geodetic line would be an elliptical arc; but if the original points were in any other circumference, the line connecting the lower extremities of the normals drawn to the earth would be a curve of double curvature. And this is the nature of the geodetic line, whatever be its position, if the earth be an irregular solid.

388. The horizontal, or azimuthal angle at any point A, between another point B on the earth's surface and the meridian AP passing through the instrument by which the angle is taken, is the inclination of the plane of that meridian to the plane passing through the said point B and through the normal to the earth's surface at A, the place of the in

h have not been exactly ascertained, a contrary result was Jained from the measurement of the lengths of a degree of ide at two places in this country; and it may be inferred ...the figure of the earth cannot be determined by geo

al operations performed in the direction of the meridian laces having little difference of latitude; even the great dional arc measured between the years 1791 and 1793, ...m Dunkirk to Barcelona, failed in this respect.

390. If the surface of the earth in the direction of the eridian of any place were nearly free from mountains and Heys the most simple and natural process for ascertaining

extent of an arc of that meridian would be that of tnting, by means of a transit instrument, a series of pickets ng the whole extent of the ground; and then determining

ngth of the line by actual measurement, applying, where essary, the corrections on account of the small inequalities the ground; that is, reducing to the curvature of the meIan, at the place, every line which is oblique to the horizon. this manner an arc of the meridian equal in extent to ut 11 degree was measured in Pennsylvania, by Mason and on, in 1764; but the inequalities of the ground, and other racles, would render such a method impracticable in most ts of the world; and therefore a recourse to other operaas would be necessary. When the length of a portion of meridian is to be ascertained, a base line from 3 to 10 Mes in length, generally on ground nearly level, is measured ; 1 convenient objects situated nearly in the direction of the ridian, at from 10 to 20 miles distance from each other hough occasionally that distance has exceeded 100 miles) ing chosen as stations, all the angles of the triangles formed the terrestrial arcs connecting those objects are taken with instrument, which, in the English operations, has always een a great theodolite. The observed angles were consetently in the plane of the horizon at each angular point. he lengths of the sides of the triangles are then calculated, nd the accuracy of the work is proved by measuring one base, r several bases, of verification as they are called: the two exemities of each base being made to serve as stations in the *riangulation, the computed length of the base is compared, and should be found to agree, with that which is obtained by the measurement. It has happened, however, that two distinct series of triangles have been carried on at once; and at, or near their terminations one line has been made common to both ties: then this line being computed independently from each

iginal bases, the agreement of the two computed ls a proof of the accuracy of all the operations.

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strument. In like manner the horizontal angle at B is the inclination of the plane of the meridian BP to the plane passing through A and the normal at B; and since the normals to the surface of a spheroid do not intersect one another unless they are drawn from points on the periphery of the same meridian or of the same parallel of latitude, it follows that the planes passing through the normals at A and B

n

m

D

cannot coincide, and that their intersections with the surface of the spheroid must form two curve lines as AmB, AnB. These being plane curves, it is evident that neither of them can be the true geodetic line detween A and B: but it has been ascertained, on setting up tall staves at two points, as A and B, in vertical positions, that is in the directions of the produced normals DA and FB, that the deviation of either staff from the vertical plane passing through the other is entirely insensible even when the most accurately adjusted instruments have been employed; consequently the planes BAD and BAF, and the curve lines Am B, Anв may, for geodetical purposes, be considered respectively as coincident.

389. One of the first admeasurements of a meridional arc in Europe, from the inaccuracy of the operations, led to an erroneous opinion respecting the figure of the earth. About the year 1718, on computing the length of an arc in the direction of the meridian between Paris and Colliure, and comparing it with the difference between the observed latitudes of its extremities, the length of the degree of latitude was found to be greater than that which was obtained in like manner from an arc between Paris and Dunkirk: and it was from thence concluded that the major axis of the terrestrial spheroid was in the direction of the poles. This result being, however, quite at variance with that which Newton had obtained from considerations founded on the effect of centrifugal force in the molecules of the earth, its justness was immediately suspected; and though the opinion of its truth was for a few years maintained in France, a repetition of the measurement led to the discovery of the particular cause of the error, and to an entirely opposite conclusion. The question was soon afterwards set at rest by a comparison of the measured lengths of a degree of the meridian at two places very distant in latitude from each other (in Lapland and Peru), which demonstrated the fact that the axis of the earth's revolution is less than a diameter of the equator. It is remarkable, however, that from causes of error

which have not been exactly ascertained, a contrary result was obtained from the measurement of the lengths of a degree of latitude at two places in this country; and it may be inferred that the figure of the earth cannot be determined by geodetical operations performed in the direction of the meridian at places having little difference of latitude; even the great meridional arc measured between the years 1791 and 1793, from Dunkirk to Barcelona, failed in this respect.

390. If the surface of the earth in the direction of the meridian of any place were nearly free from mountains and valleys the most simple and natural process for ascertaining the extent of an arc of that meridian would be that of planting, by means of a transit instrument, a series of pickets along the whole extent of the ground; and then determining the length of the line by actual measurement, applying, where necessary, the corrections on account of the small inequalities of the ground; that is, reducing to the curvature of the meridian, at the place, every line which is oblique to the horizon. In this manner an arc of the meridian equal in extent to about 1 degree was measured in Pennsylvania, by Mason and Dixon, in 1764; but the inequalities of the ground, and other obstacles, would render such a method impracticable in most parts of the world; and therefore a recourse to other operations would be necessary. When the length of a portion of

the meridian is to be ascertained, a base line from 3 to 10 miles in length, generally on ground nearly level, is measured; and convenient objects situated nearly in the direction of the meridian, at from 10 to 20 miles distance from each other (though occasionally that distance has exceeded 100 miles) being chosen as stations, all the angles of the triangles formed by the terrestrial arcs connecting those objects are taken with an instrument, which, in the English operations, has always been a great theodolite. The observed angles were consequently in the plane of the horizon at each angular point. The lengths of the sides of the triangles are then calculated, and the accuracy of the work is proved by measuring one base, or several bases, of verification as they are called: the two extremities of each base being made to serve as stations in the triangulation, the computed length of the base is compared, and should be found to agree, with that which is obtained by the measurement. It has happened, however, that two distinct series of triangles have been carried on at once; and at, or near their terminations one line has been made common to both series: then this line being computed independently from each of the original bases, the agreement of the two computed values affords a proof of the accuracy of all the operations.

A A